Justify the elementary row operations in class.
Definition: The pivot positions are positions that contain a leading term. The pivot columns are columns that contain a pivot position. A pivot is the value of a pivot position.
Algorithm: Gaussian elimination is performed as follows: * find the pivot position in the first row * use elementary row operators to eliminate all value under the pivot position * continue
work out example in class
Definition: A matrix is in reduced echelon form if * it is in echelon form * all pivot positions contain a 1 * the only nonzero term in a pivot colum is in the pivot position
Algorithm: Gauss-Jordan elimination is performed as follows: * do Gaussian elimination * divide each row by the value of its pivot * eliminate all other values in pivot column.
work out example in class.
A linear system is homogenous if the numbers to the right of the equal sign are all zero. They always have the trivial solution
A vector is a list of number with addition and scalar multiplication defined. Given vectors u = (u1, u2, …, un) ∈ ℝn, v = (v1, v2, …, vn) ∈ ℝn of equal dimension and a scalar c ∈ ℝ, we define * addition: u + v = (u1 + v1, u2 + v2, …, un + vn), * scalar multiplication: cu = (cu1, cu2, …, cun).
go over the geometry in class. tail to tip, parallelogram
Let a, b be scalars and u, v, w ∈ ℝn. Then * u + v = v + u, * a(u + v) = au + av, * (a + b)u = au + bu, * (u + v) + w = u + (v + w), * a(bu) = (ab)u, * u + ( − u) = 0, * u + 0 = 0 + u = u, * 1u = u.
Definition: The If u1, u2, …, um are vectors and c1, c2, …, cm are scalars, then
c1u1 + c2u2 + … + cmum
is a linear combination of u1, …, um. Note that the constants can be negative or zero.
give examples in class.